The Diffeomorphism Constraint Operator in Loop Quantum Gravity

TL;DR

This paper constructs a finite triangulation diffeomorphism constraint operator in Loop Quantum Gravity, demonstrating an anomaly-free continuum limit that respects the Lie algebra of diffeomorphisms, with novel features involving holonomies and fluxes.

Contribution

It introduces a new construction of the diffeomorphism constraint operator in LQG that is anomaly-free and incorporates fluxes and edge labels, advancing the understanding of quantum dynamics.

Findings

Anomaly-free continuum limit of the diffeomorphism constraint operator.

Inclusion of fluxes and edge labels in the holonomy approximants.

Insights into the 'mu-bar' scheme analogy with Loop Quantum Cosmology.

Abstract

We construct the smeared diffeomorphism constraint operator at finite triangulation from the basic holonomy- flux operators of Loop Quantum Gravity, evaluate its continuum limit on the Lewandowski- Marolf habitat and show that the action of the continuum operator provides an anomaly free representation of the Lie algebra of diffeomorphisms of the 3- manifold. Key features of our analysis include: (i) finite triangulation approximants to the curvature, $F_{ab}^i$ of the Ashtekar- Barbero connection which involve not only small loop holonomies but also small surface fluxes as well as an explicit dependence on the edge labels of the spin network being acted on (ii) the dependence of the small loop underlying the holonomy on both the direction and magnitude of the shift vector field (iii) continuum constraint operators which do {\em not} have finite action on the kinematic Hilbert space, thus implementing a key lesson from recent studies of parameterised field theory by the authors. Features (i) and (ii) provide the first hints in LQG of a conceptual similarity with the so called "mu- bar" scheme of Loop Quantum Cosmology. We expect our work to be of use in the construction of an anomaly free quantum dynamics for LQG.